265 research outputs found

    Small grid embeddings of 3-polytopes

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    We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by O(27.55n)=O(188n)O(2^{7.55n})=O(188^{n}). If the graph contains a triangle we can bound the integer coordinates by O(24.82n)O(2^{4.82n}). If the graph contains a quadrilateral we can bound the integer coordinates by O(25.46n)O(2^{5.46n}). The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

    Embedding Stacked Polytopes on a Polynomial-Size Grid

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    A stacking operation adds a dd-simplex on top of a facet of a simplicial dd-polytope while maintaining the convexity of the polytope. A stacked dd-polytope is a polytope that is obtained from a dd-simplex and a series of stacking operations. We show that for a fixed dd every stacked dd-polytope with nn vertices can be realized with nonnegative integer coordinates. The coordinates are bounded by O(n2log(2d))O(n^{2\log(2d)}), except for one axis, where the coordinates are bounded by O(n3log(2d))O(n^{3\log(2d)}). The described realization can be computed with an easy algorithm. The realization of the polytopes is obtained with a lifting technique which produces an embedding on a large grid. We establish a rounding scheme that places the vertices on a sparser grid, while maintaining the convexity of the embedding.Comment: 22 pages, 10 Figure

    Strictly convex drawings of planar graphs

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    Every three-connected planar graph with n vertices has a drawing on an O(n^2) x O(n^2) grid in which all faces are strictly convex polygons. These drawings are obtained by perturbing (not strictly) convex drawings on O(n) x O(n) grids. More generally, a strictly convex drawing exists on a grid of size O(W) x O(n^4/W), for any choice of a parameter W in the range n<W<n^2. Tighter bounds are obtained when the faces have fewer sides. In the proof, we derive an explicit lower bound on the number of primitive vectors in a triangle.Comment: 20 pages, 13 figures. to be published in Documenta Mathematica. The revision includes numerous small additions, corrections, and improvements, in particular: - a discussion of the constants in the O-notation, after the statement of thm.1. - a different set-up and clarification of the case distinction for Lemma

    Embedding Stacked Polytopes on a Polynomial-Size Grid

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    We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n4) x O(n4) x O(n18). We use a perturbation technique to construct an integral 2D embedding that lifts to a small 3D polytope, all in linear time. This result solves a question posed by G unter M. Ziegler, and is the rst nontrivial subexponential upper bound on the long-standing open question of the grid size necessary to embed arbitrary convex polyhedra, that is, about effcient versions of Steinitz's 1916 theorem. An immediate consequence of our result is that O(log n)-bit coordinates suffice for a greedy routing strategy in planar 3-trees.Deutsche Forschungsgemeinschaft (DFG) (Grant No. SCHU 2458/1-1

    Convexity-Increasing Morphs of Planar Graphs

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    We study the problem of convexifying drawings of planar graphs. Given any planar straight-line drawing of an internally 3-connected graph, we show how to morph the drawing to one with strictly convex faces while maintaining planarity at all times. Our morph is convexity-increasing, meaning that once an angle is convex, it remains convex. We give an efficient algorithm that constructs such a morph as a composition of a linear number of steps where each step either moves vertices along horizontal lines or moves vertices along vertical lines. Moreover, we show that a linear number of steps is worst-case optimal. To obtain our result, we use a well-known technique by Hong and Nagamochi for finding redrawings with convex faces while preserving y-coordinates. Using a variant of Tutte's graph drawing algorithm, we obtain a new proof of Hong and Nagamochi's result which comes with a better running time. This is of independent interest, as Hong and Nagamochi's technique serves as a building block in existing morphing algorithms.Comment: Preliminary version in Proc. WG 201

    A Quantitative Steinitz Theorem for Plane Triangulations

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    We give a new proof of Steinitz's classical theorem in the case of plane triangulations, which allows us to obtain a new general bound on the grid size of the simplicial polytope realizing a given triangulation, subexponential in a number of special cases. Formally, we prove that every plane triangulation GG with nn vertices can be embedded in R2\mathbb{R}^2 in such a way that it is the vertical projection of a convex polyhedral surface. We show that the vertices of this surface may be placed in a 4n3×8n5×ζ(n)4n^3 \times 8n^5 \times \zeta(n) integer grid, where ζ(n)(500n8)τ(G)\zeta(n) \leq (500 n^8)^{\tau(G)} and τ(G)\tau(G) denotes the shedding diameter of GG, a quantity defined in the paper.Comment: 25 pages, 6 postscript figure
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